We study two classes of extension problems, and their interconnections: (i) Extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; (ii) In case of Lie groups, representations of the associated Lie algebras $La\left(G\right)$ by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space (RKHS) $\mathscr{H}_{F}$.
Why extensions? In science, experimentalists frequently gather spectral data in cases when the observed data is limited, for example limited by the precision of instruments; or on account of a variety of other limiting external factors. Given this fact of life, it is both an art and a science to still produce solid conclusions from restricted or limited data. In a general sense, our monograph deals with the mathematics of extending some such given partial data-sets obtained from experiments. More specifically, we are concerned with the problems of extending available partial information, obtained, for example, from sampling. In our case, the limited information is a restriction, and the extension in turn is the full positive definite function (in a dual variable); so an extension if available will be an everywhere defined generating function for the exact probability distribution which reflects the data; if it were fully available. Such extensions of local information (in the form of positive definite functions) will in turn furnish us with spectral information. In this form, the problem becomes an operator extension problem, referring to operators in a suitable reproducing kernel Hilbert spaces (RKHS). In our presentation we have stressed hands-on-examples. Extensions are almost never unique, and so we deal with both the question of existence, and if there are extensions, how they relate back to the initial completion problem.
Operator systems and positive extensions over discrete groups
National and Kapodistrian University of Athens（雅典国立卡波迪斯特里安大学）；Max Planck Institute for Mathematics（马克斯·普朗克数学研究所）；University of Delaware（特拉华大学）；Radboud University Nijmegen（奈梅亨拉德堡德大学）
Universality of kernels on Riemannian symmetric spaces
Laboratoire Jean-Kuntzmann (CNRS), Université Grenoble-Alpes（让·孔茨曼实验室（法国国家科学研究中心），格勒诺布尔阿尔卑斯大学）；Tübingen AI Center, University of Tübingen（蒂宾根人工智能中心，蒂宾根大学）；Division of Mathematical Sciences, Nanyang Technological University（南洋理工大学数理学院）
The Riesz transform for the harmonic oscillator in spherical coordinates
本文给出了球面坐标中谐振子里斯变换在混合范数空间中的加权估计。为了证明这一结果，我们需要一个加权不等式的里斯变换的向量值扩展的Laguerre展开，这是独立的利益。获得这种扩展的主要工具是独立于所涉及的Laguerre函数的阶数的里斯变换的加权不等式和适当的Rubio de Francia的外推定理的适应。
